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An explicit formula for the Hilbert symbol of a formal group. (English) Zbl 1270.11122
Summary: A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by V. A. Abrashkin [Izv. Math. 61, No. 3, 463–515 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 3, 3–56 (1997; Zbl 0889.11041)] under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of \((\phi ,\Gamma)\)-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build \((\phi,\Gamma)\)-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the cup-product and the Kummer map.

MSC:
11S31 Class field theory; \(p\)-adic formal groups
11S25 Galois cohomology
14L05 Formal groups, \(p\)-divisible groups
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